To earn Battle Royale, there must be only one crew member currently employed onboard the player's ship. That sole crew member must then be teleported over and strike the final blow against the last person on the enemy ship.
This achievement cannot be obtained by killing all the crew on the flagship, likely because the final kill must trigger a victory, and the flagship has an AI that takes over.
It can also be easily unlocked in the first sector by boarding and bringing the enemy to almost no health, teleporting your crew back and heal them. Then you put one sole crew member in the teleporter and kill your other crew members with asphyxiation. Finish the last enemy off with your remaining crew member.
If playing with advanced edition enabled, and if you have a clone bay, this is an easy achievement to earn. All you need is a Backup DNA bank augmentation to make things easier. Because the game believes crew in the cloning queue are dead, if you have this augment, turn off your clone bay once all other members of your crew that WON'T be going over are dead, send over your remaining crew member to deal with the enemy ship. This also means that you aren't throwing away your current run by going for this achievement. Once you are finished, just power up your clone bay and all of you crew will be ready to go back to their positions (with a 20% skill decrease though).